Modeling 3-D anisotropic fractal mediaa

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Chemingui 8 Fractal media v=1.0 v=0.5 v=0.4 v=0.3 v=0.2 v=0.1 v=0.0 Figure 5: Synthetic random sequences simulating acoustic impedances with von Karman autocorrelation for varying Hurst number ν. describe a random field in which the medium can be represented as a two-state model. This new field is called a binary field and the process of deriving the binary field from the continuous field is called “binarization” (Holliger et al., 1993). The problem is to relate the statistics of the binary field to those of the continuous field. Holliger et al. (1993) gave a brief description of their mapped two-dimensional binary field which I apply in a straightforward generalization to the three-dimensional problem. To illustrate the effects of “binarizing” a continuous field, let’s consider two examples of random fields with Gaussian and exponential autocorrelation functions, respectively. In the first example I simulate a randomly-stratified medium. The second example is a realization of a random medium with statistically isotropic homogeneous inclusions. I like to analyze the change in the medium properties by comparing the autocorrelation function of the distribution before and after “binarization”. For better observation, I limit the analysis to the study of the correlation function along one axis, i.e, in the x-direction. Figure ?? shows the averaged 1-D correlation function along the x-axis for the randomly layered medium. The solid curve displays the autocorrelation of the continuous field; the dashed one represents the autocorrelation of the “binary” field. The two functions are noticeably different from one another; the slope near the origin is greatly increased after “binarization” indicating a rougher distribution compared to the continuous case. Figure ?? shows the same observations for the isotropic random field with Gaussian autocorrelation; again the roughness of the field has increased as SEP–80
Chemingui 9 Fractal media indicated by the steepening in the slope of the autocorrelation. Figure 6: Synthetic continuous random field with apparent layering and Gaussian autocorrelation; a x = 5, a y = 80, a z = 80. CONCLUSIONS In this initial study I have tackled the forward problem for modeling anisotropic fractal media using second-order statistics. The method has close analogy with the two-dimensional Goff and Jordan model for seafloor morphology. The generation of synthetic models is done in the Fourier domain and the algorithms are similar for the one- two- and three-dimensional problems. The von Karman functions are presented as a generalization of the exponential correlation function associated with the Markov process in modeling seismic impedances. The von Karman functions can be used for better description of statistic lithology of stratigraphic columns and understanding their depositional pattern. I have also computed a two-state model (i.e., rock/pore or sandstone/shale) by mapping the random field from continuous realizations into a binary field. Comparisons of the autocorrelation functions of the continuous and binary fields show that the fractal dimension (i.e, the roughness of the medium) increases through the “binarization” process. FUTURE WORK Future goals of this effort will be to formulate the inverse problem for estimating the characteristic parameters of the anisotropic fractal medium, i.e, aspect ratios of anisotropy, and Hausdorff (fractal) dimension. The technique of deriving the binary SEP–80
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Modeling 3-D anisotropic fractal mediaa

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